From 0b582ec3a43c69fb09104b6f3ed4dfdf037f48fa Mon Sep 17 00:00:00 2001 From: Mark Date: Thu, 10 Oct 2024 14:32:58 -0700 Subject: [PATCH] Added masses intro --- Advanced/Geometry of Masses/main.tex | 34 ++++ .../Geometry of Masses/parts/0 balance 1d.tex | 176 ++++++++++++++++++ .../Geometry of Masses/parts/1 balance 2d.tex | 148 +++++++++++++++ 3 files changed, 358 insertions(+) create mode 100755 Advanced/Geometry of Masses/main.tex create mode 100644 Advanced/Geometry of Masses/parts/0 balance 1d.tex create mode 100644 Advanced/Geometry of Masses/parts/1 balance 2d.tex diff --git a/Advanced/Geometry of Masses/main.tex b/Advanced/Geometry of Masses/main.tex new file mode 100755 index 0000000..04c0e56 --- /dev/null +++ b/Advanced/Geometry of Masses/main.tex @@ -0,0 +1,34 @@ +% use [nosolutions] flag to hide solutions. +% use [solutions] flag to show solutions. +\documentclass[ + solutions, + singlenumbering, + shortwarning +]{../../resources/ormc_handout} +\usepackage{../../resources/macros} + +\usepackage{tikz} +\usetikzlibrary{patterns} +\usetikzlibrary{shapes.geometric} + +\usepackage{graphicx} + + + +\uptitlel{Advanced 2} +\uptitler{\smallurl{}} +\title{Geometry of Masses I} +\subtitle{Prepared by Sunny \& Mark on \today{}} + + + + +\begin{document} + \maketitle + + \input{parts/0 balance 1d.tex} + \input{parts/1 balance 2d.tex} + %\input{parts/1 continuous} + %\input{parts/2 pappus} + +\end{document} \ No newline at end of file diff --git a/Advanced/Geometry of Masses/parts/0 balance 1d.tex b/Advanced/Geometry of Masses/parts/0 balance 1d.tex new file mode 100644 index 0000000..f2172bb --- /dev/null +++ b/Advanced/Geometry of Masses/parts/0 balance 1d.tex @@ -0,0 +1,176 @@ +\section{Balancing a line} + +\example{} +Consider a mass $m_1$ on top of a pin. \par +Due to gravity, the mass exerts a force on the pin at the point of contact. \par +For simplicity, we'll say that the magnitude of this force is equal the mass of the object--- +that is, $m_1$. +\begin{center} + \begin{tikzpicture}[scale=2] + \fill[color = black] (0, 0.1) circle[radius=0.1]; + \node[above] at (0, 0.20) {$m_1$}; + + \draw[line width = 0.25mm, pattern=north west lines] (0, 0) -- (-0.15, -0.3) -- (0.15, -0.3) -- cycle; + + + \draw[color = black, opacity = 0.5] (1, 0.1) circle[radius=0.1]; + \draw[line width = 0.25mm, pattern=north west lines, opacity = 0.5] (1, 0) -- (0.85, -0.3) -- (1.15, -0.3) -- cycle; + + \draw[->, line width = 0.5mm] (1, 0) -- (1, -0.5) node[below] {$m_1$}; + %\draw[->, line width = 0.5mm, dashed] (1, 0) -- (1, 0.5) node[above] {$-m_1$}; + + \fill[color = red] (1, 0) circle[radius=0.025]; + \end{tikzpicture} +\end{center} +The pin exerts an opposing force on the mass at the same point, and the system thus stays still. + + +\remark{} +Forces, distances, and torques in this handout will be provided in arbitrary (though consistent) units. \par +We have no need for physical units in this handout. + +\example{} +Now attach this mass to a massless rod and try to balance the resulting system. \par +As you might expect, it is not stable: the rod pivots and falls down. +\begin{center} + \begin{tikzpicture}[scale=2] + \fill[color = black] (-0.3, 0.0) circle[radius=0.1]; + \node[above] at (-0.3, 0.1) {$m_1$}; + \draw[-, line width = 0.5mm] (-0.8, 0) -- (0.5, 0); + + \draw[line width = 0.25mm, pattern=north west lines] (0, 0) -- (-0.15, -0.3) -- (0.15, -0.3) -- cycle; + + + \draw[color = black, opacity = 0.5] (1.2, 0.0) circle[radius=0.1]; + \draw[-, line width = 0.5mm, opacity = 0.5] (0.7, 0) -- (1.9, 0); + + \draw[line width = 0.25mm, pattern=north west lines, opacity = 0.5] (1.5, 0) -- (1.35, -0.3) -- (1.65, -0.3) -- cycle; + + + \draw[->, line width = 0.5mm] (1.2, 0) -- (1.2, -0.5) node[below] {$m_1$}; + %\draw[->, line width = 0.5mm, dashed] (1.5, 0) -- (1.5, 0.5) node[above] {$f_p$}; + \end{tikzpicture} +\end{center} +This is because the force $m_1$ is offset from the pivot (i.e, the tip of the pin). \par +It therefore exerts a \textit{torque} on the mass-rod system, causing it to rotate and fall. + +\pagebreak + + +\definition{Torque} +Consider a rod on a single pivot point. +If a force with magnitude $m_1$ is applied at an offset $d$ from the pivot point, +the system experiences a \textit{torque} with magnitude $m_1 \times d$. +\begin{center} + \begin{tikzpicture}[scale=2] + \draw[-, line width = 0.5mm] (-1.2, 0) -- (0.5, 0); + \draw[line width = 0.25mm, pattern=north west lines] (0, 0) -- (-0.15, -0.3) -- (0.15, -0.3) -- cycle; + + \draw[->, line width = 0.5mm, dashed] (-0.8, 0) -- (-0.8, -0.5) node[below] {$m_1$}; + \fill[color = red] (-0.8, 0.0) circle[radius=0.05]; + + + \draw[-, line width = 0.3mm, double] (-0.8, 0.1) -- (-0.8, 0.2) -- (0, 0.2) node [midway, above] {$d$} -- (0, 0.1); + \end{tikzpicture} +\end{center} + +We'll say that a \textit{positive torque} results in \textit{clockwise} rotation, +and a \textit{negative torque} results in a \textit{counterclockwise rotation}. +As stated in \ref{fakeunits}, torque is given in arbitrary \say{torque units} +consistent with our units of distance and force. + +\vspace{2mm} + +% I believe the convention used in physics is opposite ours, but that's fine. +% Positive = clockwise is more intuitive given our setup, +% and we only use torque to define CoM anyway. +Look at the diagram above and convince yourself that this convention makes sense: +\begin{itemize} + \item $m_1$ is positive \note{(masses are usually positive)} + \item $d$ is negative \note{($m_1$ is \textit{behind} the pivot)} + \item therefore, $m_1 \times d$ is negative. +\end{itemize} + + +\definition{Center of mass} +The \textit{center of mass} of a physical system is the point at which one can place a pivot \par +so that the total torque the system experiences is 0. \par +\note{In other words, it is the point at which the system may be balanced on a pin.} + + +\problem{} +Consider the following physical system: +we have a massless rod of length $1$, with a mass of size 3 at position $0$ +and a mass of size $1$ at position $1$. +Find the position of this system's center of mass. \par + +\begin{center} + \begin{tikzpicture}[scale=2] + \draw[line width = 0.25mm, pattern=north west lines] (0, 0) -- (-0.15, -0.3) -- (0.15, -0.3) -- cycle; + + \draw[-, line width = 0.5mm] (-0.5, 0) -- (1.5, 0); + + + \fill[color = black] (-0.5, 0) circle[radius=0.1]; + \node[above] at (-0.5, 0.2) {$3$}; + + \fill[color = black] (1.5, 0) circle[radius=0.08]; + \node[above] at (1.5, 0.2) {$1$}; + \end{tikzpicture} +\end{center} + +\vfill + +\problem{} +Do the same for the following system, where $m_1$ and $m_2$ are arbitrary masses. + +\begin{center} + \begin{tikzpicture}[scale=2] + \draw[line width = 0.25mm, pattern=north west lines] (0.7, 0) -- (0.55, -0.3) -- (0.85, -0.3) -- cycle; + + \draw[-, line width = 0.5mm] (-0.5, 0) -- (1.5, 0); + + + \fill[color = black] (-0.5, 0) circle[radius=0.1]; + \node[above] at (-0.5, 0.2) {$m_1$}; + + \fill[color = black] (1.5, 0) circle[radius=0.08]; + \node[above] at (1.5, 0.2) {$m_2$}; + \end{tikzpicture} +\end{center} + +\vfill +\pagebreak + +\definition{} +Consider a massless, horizontal rod of infinite length. \par +Affix a finite number of point masses to this rod. \par +We will call the resulting object a \textit{one-dimensional system of masses}: +\begin{center} + \begin{tikzpicture}[scale=1] + \draw[<->, line width = 0.5mm] (-4, 0) -- (4, 0); + \node[left] at (-4, 0) {$...$}; + \node[right] at (4, 0) {$...$}; + + + \fill[color = black] (-2.5, 0) circle[radius=0.12]; + \node[above] at (-2.5, 0.15) {$m_1$}; + + \fill[color = black] (-0.5, 0) circle[radius=0.1]; + \node[above] at (-0.5, 0.15) {$m_2$}; + + \fill[color = black] (1.5, 0) circle[radius=0.15]; + \node[above] at (1.5, 0.15) {$m_3$}; + \end{tikzpicture} +\end{center} +\vspace{5mm} + + + +\problem{} +Consider a one-dimensional system of masses consisting of $n$ masses $m_1, m_2, ..., m_n$, \par +with each $m_i$ positioned at $x_i$. Show that the resulting system always has a unique center of mass. \par +\hint{Prove this by construction: find the point!} + +\vfill +\pagebreak \ No newline at end of file diff --git a/Advanced/Geometry of Masses/parts/1 balance 2d.tex b/Advanced/Geometry of Masses/parts/1 balance 2d.tex new file mode 100644 index 0000000..8bada31 --- /dev/null +++ b/Advanced/Geometry of Masses/parts/1 balance 2d.tex @@ -0,0 +1,148 @@ +\section{Balancing a plane} + +\definition{} +Consider a massless two-dimensional plane. \par +Affix a finite number of point masses to this plane. \par +We will call the resulting object a \textit{two-dimensional system of masses:} + + +\begin{center} + \begin{tikzpicture}[scale = 0.5] + + %\draw[ + % line width = 0mm, + % pattern = north west lines, + % pattern color = blue, + %] + % (1, 0) + % -- (0.5, 0.866) + % -- (-0.5, 0.866) + % -- (-1, 0) + % -- (-0.5, -0.866) + % -- (0.5, -0.866) + % -- cycle; + %\draw[ + % line width = 0.5mm, + % blue + %] + % (1, 0) + % -- (0.5, 0.866) + % -- (-0.5, 0.866) + % -- (-1, 0) + % -- (-0.5, -0.866) + % -- (0.5, -0.866) + % -- cycle; + %\fill[color = blue] (0, 0) circle[radius=0.3]; + + + \fill[color = black] + (-3, 3) circle[radius = 0.5] + node[above] at (-3, 3.5) {$m_1$ at $(x_1, y_1)$}; + + \fill[color = black] + (-5, -1.5) circle[radius = 0.4] + node[above] at (-5, -1.0) {$m_2$ at $(x_2, y_2)$}; + + \fill[color = black] + (3, -3) circle[radius = 0.35] + node[above] at (3, -2.5) {$m_3$ at $(x_3, y_3)$}; + + \draw[line width = 0.5mm] + (-7.5, -4.2) + -- (6, -4.2) + -- (6, 5) + -- (-7.5, 5) + -- cycle; + + \end{tikzpicture} +\end{center} +\vspace{5mm} + + + +\problem{} +Show that any two-dimensional system of masses has a unique center of mass. \par +\hint{ + If a plane balances on a pin, it does not tilt in the $x$ or $y$ direction. \par + See the diagram below. +} +\begin{center} + \begin{tikzpicture}[scale = 0.5] + + % Horizontal + \draw[line width = 0.5mm, dotted, gray] (-3, 3) -- (-3, -5); + \draw[line width = 0.5mm, dotted, gray] (-5, -1.5) -- (-5, -5); + \draw[line width = 0.5mm, dotted, gray] (3, -3) -- (3, -5); + \draw[line width = 0.5mm, dotted, gray] (0, 0) -- (0, -5); + \draw[line width = 0.5mm] (-7, -5) -- (6.5, -5); + + \fill[color = gray] (-3, -5) circle[radius = 0.3]; + \fill[color = gray] (-5, -5) circle[radius = 0.3]; + \fill[color = gray] (3, -5) circle[radius = 0.3]; + + \draw[line width = 0.25mm, pattern=north west lines] + (0, -5) -- (-0.6, -6) -- (0.6, -6) -- cycle; + + + % Vertical + + \draw[line width = 0.5mm, dotted, gray] (-3, 3) -- (8, 3); + \draw[line width = 0.5mm, dotted, gray] (-5, -1.5) -- (8, -1.5); + \draw[line width = 0.5mm, dotted, gray] (3, -3) -- (8, -3); + \draw[line width = 0.5mm, dotted, gray] (0, 0) -- (8, 0); + \draw[line width = 0.5mm] (8, 4) -- (8, -4); + + \fill[color = gray] (8, 3) circle[radius = 0.3]; + \fill[color = gray] (8, -1.5) circle[radius = 0.3]; + \fill[color = gray] (8, -3) circle[radius = 0.3]; + + \draw[line width = 0.25mm, pattern=north west lines] + (8, 0) -- (9, -0.6) -- (9, 0.6) -- cycle; + + + \draw[ + line width = 0mm, + pattern = north west lines, + pattern color = blue, + ] + (1, 0) + -- (0.5, 0.866) + -- (-0.5, 0.866) + -- (-1, 0) + -- (-0.5, -0.866) + -- (0.5, -0.866) + -- cycle; + \draw[ + line width = 0.5mm, + blue + ] + (1, 0) + -- (0.5, 0.866) + -- (-0.5, 0.866) + -- (-1, 0) + -- (-0.5, -0.866) + -- (0.5, -0.866) + -- cycle; + \fill[color = blue] (0, 0) circle[radius=0.3] + node[above] at (0, 1) {Pivot}; + + \fill[color = black] + (-3, 3) circle[radius = 0.5] + node[above] at (-3, 3.5) {$m_1$ at $(x_1, y_1)$}; + + \fill[color = black] + (-5, -1.5) circle[radius = 0.4] + node[above] at (-5.5, -1.0) {$m_2$ at $(x_2, y_2)$}; + + \fill[color = black] + (3, -3) circle[radius = 0.35] + node[above] at (3, -2.8) {$m_3$ at $(x_3, y_3)$}; + \end{tikzpicture} +\end{center} + + +\vfill +\pagebreak + +\vfill +\pagebreak \ No newline at end of file